When the dimensions of a metallic conductor are reduced so that they become comparable to the de Broglie wavelengths of the conduction electrons, the absence of scattering results in ballistic electron transport(1) and the conductance becomes quantized(2-4). In ferromagnetic metals, the spin angular momentum of the electrons results in spin-dependent conductance quantization(5-7) and various unusual magnetoresistive phenomena(8-12). Theorists have predicted a related phenomenon known as ballistic anisotropic magnetoresistance (BAMR)(13). Here we report the first experimental evidence for BAMR by observing a stepwise variation in the ballistic conductance of cobalt nanocontacts as the direction of an applied magnetic field is varied. Our results show that BAMR can be positive and negative, and exhibits symmetric and asymmetric angular dependences, consistent with theoretical predictions.